Survival analysis in decision modeling

petros.pechlivanoglou@sickkids.ca

• Motivation and introduction in Survival analysis
• Survival modeling
• Partitioned survival modeling
• Multistate models
• Advanced topiics
  • Mixture cure models
  • External evidence in survival models

• RCT evidence often not sufficient for Health Technology Assesment (HTA)
  • Short follow up
  • Patients lost during follow up
  • Often not reflective of real world
  • Right skewed
    
  • Information “flowing” over intervals
  • Extrapolation often necessary

• Overall Survival (OS), Progression-Free Survival (PFS)
• Trials powered to show differences on PFS and not OS
• Cross-over
• PFS and OS data are correlated
• Access to patient level OS data not always the case

• Set of statistical methods that can handle:

  • Data skewness
  • Censoring
  • (Time-dependent) Covariates
  • State transition processes
  • Recurrent events
  • The expectation of cure

• Used for both inference and extrapolation

• Survival: the probability of not experiencing an event unit some time \( t \)

  \( S(t) = Pr(T >t) = 1- Pr(T \leq t) \)

• Non-parametric models (e.g. Kaplan Meier)
• Semi-Parametric models (e.g. Cox Proportional Hazard)
• Parametric models (e.g. Accelerated Failure Time)

• Cox proportional hazard model
  • Hazard: the instantaneous risk of an event at time \( t \), conditional on survival to that time.
• Semi:
  • Baseline hazard modelled non parametrically (i.e effect of time)
  • Covariate effects ( \( \beta \) 's) modelled parametrically (like in a regression model)

• Usually model time-to-event directly rather than hazard
• Resemble a regression model but can capture censoring
• Fit a distribution to the (observed) time-to-event data
  • 1 parameter : exponential
  • 2 parameter : Weibull, lognormal, gamma etc
  • 3, 4 parameter: Gen. gamma , Gen F etc
• Assumption that censored patients will follow similar patterns to the observed.
• Temporal extrapolation possible !

• Statistical considerations of goodness-of-fit
• Clinical plausibility
  • in-sample
  • extrapolations
  • out-of-sample

• Likelihood based metrics

  • Akaike Information Criterion (AIC)
  • Bayesian Information Criterion (BIC)

• Performance based metrics

  • Sensitivity , Specificity
  • (time dependent) Receiver Operating characteristic (ROC) curve
  • Area Under the Curve (AUC)
  • c- statistic

• NICE Decision Support Unit
  • Latimer et al 2011 : Survival Analysis For Economic Evaluations Alongside Clinical Trials - Extrapolation with Patient-Level Data
  • Survival Model Selection Process algorithm:
    • recommendations for how survival analysis can be undertaken more systematically. This involves fitting and testing a range of survival models and comparing these based upon internal validity (how well they fit to the observed trial data) and external validity (how plausible their extrapolated portions are)

Survival models where the underlying hazard of an event is modelled as a smooth, piecewise polynomial function of time.

• Extrapolation of hazard possible
• Non-linear relation of time and hazard
• Conventionally knots are evenly spread on the (log)-time axis
• Overfitting with too many knots possible -sensitivity analysis
• AIC, BIC valid GoF tests

Gibson et al (Pharmacoeconomics, 2017)

From Gibson et al (Pharacoeconomcs, 2017)

Spline-based models using a limited number of knots can provide an acceptable fit to trial data and generate extrapolated estimates supported by longer term evidence, with results that are stable in response to changes in knot placement.

Modeler faced with a more decisions:

• Extrapolate using KM and “fitted” tails

• Extrapolate using the fitted curve

• Fit seperate curves to the data

• Extrapolate treatment effect as relative

  • What is the behaviour of the relative effect post trial?

• Intuitively appealing
• Easy to communicate
• Easy to construct
• In-sync with what is commonly reported in RCTs
• Can be constructed using aggregate / graphic based data
• In-sync with methods of cross-over
• PSMs work great for cases where the whole cohort is observed until event of interest (unlikely in the vast majority of the cases)

• OS and PFS are falsely assumed independent
• Cohort cannot transition back to a healthier state.
• 3-state PSM cannot distinguish the origin of the cohort moving to the dead state (progresed -> death vs preprogession -> death)
• In the presence of censoring:
  • Projected trajectory of state occupancy after end of follow up only informed by the observed trajectory
  • Poor performance when trends in the within trial period may not continue in the extrapolation period

• Extended form of competing risks
• Multivariate survival analysis

• Can incorporate:

  • Transition specific covariates
  • Recurrent events

• Can work with

  • Patient-level data (best)
  • Digitized / interval censored data (…not best)

Fitted in two ways:

1. separate models for each trasition
   • recurrent events
   • covariates
   • seperate dataset for each (non-saturating) state
   • “naive” assumption of censoring when competing event occurs
   • easy to fit in R flexsurv and mstate

Fitted in two ways:

1. Joint multivariate model for all transitions - Powerfull!
   • Mislclassification errors
   • Latent states
   • Interval censoring
   • Continuous time

Drawbacks:

• Difficult to converge
• Limited options wrt assumed distributions (exponential)

Can be fitted through msm and flexsurv

Given short term follow up in RCTs, survival extrapolations can be informed by external data

• Adjustment of all-cause mortality using life-tables
• Expert opinion on plausibility

Other sources of external data:

• longer follow-up on a different RCT
• disease registry
• administrative data

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• Therapies with the possibility of cure pose a modeling challenge
• e.g. plateau on immotherapy survival curves
• Extrapolation challenging without external information
• Traditional methods: Underestimation more likely than over estimation of the effect

• Cure rate models a promising solution but the cure faction difficult to estimate

• Can be fitted in both R and SAS

• Guidance is lacking but given the popularity soon to come!